Solution of Multibody Dynamics Using Natural Factors and Iterative Refinement: Part II — Closed Kinematic Loops

1989 ◽  
Author(s):  
R. A. Wehage
Keyword(s):  
Computer Architecture ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Open Loop ◽  
Force Balance ◽  
Iterative Refinement ◽  
Dynamic Force ◽  
Computational Overhead ◽  
Natural Factors ◽  
Kinematic Loops

Abstract A symbolic algorithm exploiting natural factors of generalized inertia matrices and iterative refinement to compute the dynamics of open kinematic-loop systems was developed in Part I of this paper. The general equations of motion for open and closed loop systems were derived in an earlier paper (Wehage, 1988) and it was shown that algorithms for open loop dynamics could be used to solve closed loop problems by cutting the secondary joints. In this paper it is shown that secondary joint forces can be obtained either from a dynamic force balance or from constraint surface deformations. Closed kinematic loops create additional numerical problems and require substantially more computational overhead. Therefore the iterative refinement algorithm developed in Part I is extended to address some of these problems. Exploitation of iterative refinement and computer architecture can substantially improve overall algorithm performance.

Analytical Modelling of the Dynamics of the Four-Bar Mechanism: A Comparison of Some Methods

2018 ◽  
Author(s):  
J. P. Meijaard ◽  
V. van der Wijk
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Force Balance ◽  
Dynamic Force ◽  
Algebraic Equations ◽  
Principle Of Virtual Work ◽  
Principal Vector ◽  
Reaction Forces ◽  
Principal Points

Some thoughts about different ways of formulating the equations of motion of a four-bar mechanism are communicated. Four analytic methods to derive the equations of motion are compared. In the first method, Lagrange’s equations in the traditional form are used, and in a second method, the principle of virtual work is used, which leads to equivalent equations. In the third method, the loop is opened, principal points and a principal vector linkage are introduced, and the equations are formulated in terms of these principal vectors, which leads, with the introduced reaction forces, to a system of differential-algebraic equations. In the fourth method, equivalent masses are introduced, which leads to a simpler system of principal points and principal vectors. By considering the links as pseudorigid bodies that can have a uniform planar dilatation, a compact form of the equations of motion is obtained. The conditions for dynamic force balance become almost trivial. Also the equations for the resulting reaction moment are considered for all four methods.

A Computational Method for the Dynamic Analysis of Planar Linkages With Applications

2001 ◽  
Author(s):  
Hazem A. Attia
Keyword(s):  
Closed Loop ◽  
Equations Of Motion ◽  
Open Loop ◽  
Rigid Bodies ◽  
Transformation Matrix ◽  
Computational Method ◽  
Constraint Forces ◽  
Constrained System ◽  
Velocity Transformation ◽  
Planar Linkages

Abstract This paper presents a computational method for generating the equations of motion of planar linkages consisting of interconnected rigid bodies. The formulation uses initially the rectangular Cartesian coordinates of an equivalent constrained system of particles to define the configuration of the mechanism. This results in a simple and straightforward procedure for generating the equations of motion. The equations of motion are then derived in terms of relative joint variables through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. Examples are used to demonstrate the generality and efficiency of the proposed method.

General Formulation of an Efficient Recursive Algorithm Based on Canonical Momenta for Forward Dynamics of Closed-Loop Multibody Systems

2005 ◽  
Author(s):  
Joris Naudet ◽  
Dirk Lefeber
Keyword(s):  
Multibody Systems ◽  
Closed Loop ◽  
Recursive Algorithm ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Open Loop ◽  
Forward Dynamics ◽  
Virtual Power ◽  
Generalized Coordinates ◽  
First Order

In previous work, a method for establishing the equations of motion of open-loop multibody mechanisms was introduced. The proposed forward dynamics formulation resulted in a Hamiltonian set of 2n first order ODE’s in the generalized coordinates q and the canonical momenta p. These Hamiltonian equations were derived from a recursive Newton-Euler formulation. It was shown how an O(n) formulation could be obtained in the case of a serial structure with general joints. The amount of required arithmetical operations was considerably less than comparable acceleration based formulations. In this paper, a further step is taken: the method is extended to constrained multibody systems. Using the principle of virtual power, it is possible to obtain a recursive Hamiltonian formulation for closed-loop mechanisms as well, enabling the combination of the low amount of arithmetical operations and a better evolution of the constraints violation errors, when compared with acceleration based methods.

Solution of Multibody Dynamics Using Natural Factors and Iterative Refinement: Part I — Open Kinematic Loops

1989 ◽  
Author(s):  
R. A. Wehage
Keyword(s):  
Sparse Matrices ◽  
Equations Of Motion ◽  
Iterative Refinement ◽  
Connectivity Matrix ◽  
Recursive Algorithms ◽  
Constraint Forces ◽  
Kinematic Chains ◽  
Kinematic Loops ◽  
Free Body ◽  
Absolute Coordinates

Abstract An O(n) methodology employing block matrix partitioning and recursive projection to solve multibody equations of motion coupled by a sparse connectivity matrix was developed in (Wehage 1988, 1989, Wehage and Shabana, 1989). These primitive equations, which include all joint generalized and absolute coordinates and constraint reaction forces, are easily obtained from free body diagrams. The corresponding recursive algorithms isolate the generalized joint accelerations for numerical integration and offer the best computational advantage when solving long kinematic chains on serial processors. Recursion, however, precludes effective exploitation of vector or parallel processors. Therefore this paper explores less recursive algorithms by applying the inverse of joint connectivity to eliminate absolute accelerations and constraint forces yielding a generalized system of equations. The resulting positive definite generalized inertia matrix is first represented symbolically as a product of sparse matrices, of which some are singular and then as the product of nonsingular factors obtained recursively. This algorithm has overhead ranging from O(n2) to O(n) depending on the degree of system parallelism. Incorporating iterative refinement and exploiting parallel and vector processing makes this approach competitive for many applications.

Formulation of Multibody Dynamics Equations in Absolute or Relative Coordinates Using the Vector-Network Method

1994 ◽  
Author(s):  
John J. McPhee
Keyword(s):  
Closed Loop ◽  
Multibody System ◽  
Equations Of Motion ◽  
Open Loop ◽  
Graph Representation ◽  
Linear Graph ◽  
Relative Coordinates ◽  
Network Method ◽  
Complete Set ◽  
Theoretical Procedure

Abstract The objective of this paper is to show how a single linear graph representation of a multibody system can be used to derive the complete set of equations of motion in either absolute or relative coordinates, depending upon the elements selected into the spanning tree of the linear graph. Criteria for selecting a tree that gives the desired set of equations is given and the systematic nature of this graph-theoretical procedure, known as the Vector-Network Method, is demonstrated by means of two planar examples. The first is an open-loop compound pendulum, and the second is a closed-loop four-bar mechanism driven by a time-varying torque.

Recursive Dynamics of Overconstrained Mechanisms

1994 ◽  
Author(s):  
Udo Rein
Keyword(s):  
Closed Loop ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Open Loop ◽  
Loop Closure ◽  
Kinematic Chains ◽  
Reaction Forces ◽  
Overconstrained Mechanisms ◽  
Recursive Dynamics ◽  
Closure Constraints

Abstract Overconstrained mechanisms contain loop-closure constraints which are redundant due to a special geometry of the links. Some reaction forces of an overconstrained mechanism cannot be calculated from the dynamics of the mechanism. This means that an overconstrained mechanism is statically indeterminate. The recursive formalism was originally developed to derive the equations of motion for open-loop kinematic chains, but it has been extended by various authors to closed-loop mechanisms. This paper discusses the recursive formalism when it is applied to an overconstrained closed-loop mechanism. It will be shown that the redundant loop-closure constraints lead to rather small singular, but consistent sets of linear equations for the reaction forces at the corresponding cut joints. This means that the reaction forces at those cut joints are not unique for an overconstrained mechanism, but the variety of possible solutions does not affect the dynamics of the overconstrained mechanism. This behaviour of the recursive formalism can be used to perform an on-line investigation of the static indeterminacy of a mechanism, including singular positions, where joint constraints are redundant only at one specific position.

On closed-loop equilibrium strategies for mean-field stochastic linear quadratic problems

2020 ◽  
Vol 26 ◽  
pp. 41
Author(s):  
Tianxiao Wang
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Open Loop ◽  
Time Inconsistency ◽  
Control Problems ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Equilibrium Strategies

This article is concerned with linear quadratic optimal control problems of mean-field stochastic differential equations (MF-SDE) with deterministic coefficients. To treat the time inconsistency of the optimal control problems, linear closed-loop equilibrium strategies are introduced and characterized by variational approach. Our developed methodology drops the delicate convergence procedures in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. When the MF-SDE reduces to SDE, our Riccati system coincides with the analogue in Yong [Trans. Amer. Math. Soc. 369 (2017) 5467–5523]. However, these two systems are in general different from each other due to the conditional mean-field terms in the MF-SDE. Eventually, the comparisons with pre-committed optimal strategies, open-loop equilibrium strategies are given in details.

Comparison of Dc-Dc SEPIC, CUK and Flyback Converters Based LED Drivers

2020 ◽  
pp. 99-107
Author(s):  
Erdal Sehirli
Keyword(s):  
Integrated Circuit ◽  
Closed Loop ◽  
Input Voltage ◽  
Open Loop ◽  
Current Sensor ◽  
Switching Frequency ◽  
Led Driver ◽  
Advantages And Disadvantages ◽  
Led Drivers ◽  
Conduction Mode

This paper presents the comparison of LED driver topologies that include SEPIC, CUK and FLYBACK DC-DC converters. Both topologies are designed for 8W power and operated in discontinuous conduction mode (DCM) with 88 kHz switching frequency. Furthermore, inductors of SEPIC and CUK converters are wounded as coupled. Applications are realized by using SG3524 integrated circuit for open loop and PIC16F877 microcontroller for closed loop. Besides, ACS712 current sensor used to limit maximum LED current for closed loop applications. Finally, SEPIC, CUK and FLYBACK DC-DC LED drivers are compared with respect to LED current, LED voltage, input voltage and current. Also, advantages and disadvantages of all topologies are concluded.

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